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Helmholtz Differential Equation--Conical Coordinates

In conical coordinates, Laplace's equation can be written

(partial^2V)/(partialalpha^2)+(partial^2V)/(partialbeta^2)+(mu^2-nu^2)partial/(partiallambda)(lambda^2(partialV)/(partiallambda))=0,
(1)

where

alpha=int_a^mu(dmu)/(sqrt((mu^2-a^2)(b^2-mu^2)))
(2)
beta=int_0^nu(dnu)/(sqrt((a^2-nu^2)(b^2-nu^2)))
(3)

(Byerly 1959). Letting

V=U(u)R(r)
(4)

breaks (1) into the two equations,

d/(dr)(r^2(dR)/(dr))=m(m+1)R
(5)
(partial^2U)/(partialalpha^2)+(partial^2U)/(partialbeta^2)+m(m+1)(mu^2-nu^2)U=0.
(6)

Solving these gives

R(r)=Ar^m+Br^(-m-1)
(7)
U(u)=E_m^p(mu)E_m^p(nu),
(8)

where E_m^p are ellipsoidal harmonics of the first kind. The regular solution is therefore

V=Ar^mE_m^p(mu)E_m^p(nu).
(9)

However, because of the cylindrical symmetry, the solution E_m^p(mu)E_m^p(nu) is an mth degree spherical harmonic.

See also

Conical Coordinates, Helmholtz Differential Equation

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References

Arfken, G. "Conical Coordinates (xi_1,xi_2,xi_3)." §2.16 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 118-119, 1970.Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 263, 1959.Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 39-40, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 514 and 659, 1953.

Cite this as:

Weisstein, Eric W. "Helmholtz Differential Equation--Conical Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HelmholtzDifferentialEquationConicalCoordinates.html

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